1.Introduction

A. Bremner, AJAI CHOUDHRY and M. Ulas[1] gave a parametric solution of 2X^6 - 2Y^6 + Z^6 = W^3 as a special case of 
PX^6+ QY^6+ RZ^3+ SW^3= 0.

Noam D. Elkies[2] showed PX^6+ QY^6+ RZ^3+ SW^6= 0 has infinitely many ratinoal solutions by using the identity
(t^2+t-1)^3+(t^2-t-1)^3=2t^6-2.

We show a different parametric solution of a^2X^6 + b^2Y^6 + 2abZ^3 = W^6.


2.Theorem
        
     
    a^2X^6 + b^2Y^6 + 2abZ^3 = W^6  has infinitely many parametric solutions.
    
    a is arbitrary.
    b=1-a.
 
Proof.

a^2X^6 + b^2Y^6 + 2abZ^3 -W^6=0................................(1)

We use an identity a^2T^6 + b^2 + 2abT^3 = (aT^3+b)^2.
Let aT^3 + b = mz^3............................................(2)
Let m=a+b, b=1-a.
Substitute T=1+t, z=1+kt to equation (2).
 
v^2=-3k^4+12k^3a-18k^2a+12ka-3a^2 has a rational point (k,v)=Q(a,3a(a-1)).

Hence this quartic equation is birationally equivalent to an elliptic curve below.

Y^2+(8a-4)YX = X^3+(2a^2-2a-4)X^2+(108a^4-216a^3+108a^2)X-648a^5+216a^4+648a^3+216a^6-432a^2

This curve has point P(-2a^2+2a+4, 16a^3-24a^2-24a+16).

This point P is of infinite order, and the multiples mP, m = 2, 3, ...give infinitely many points.

Hence we can obtain infinitely many integer solutions for equation (1).

Case : m=2

X = (a-2)(a^4+10a^3-12a^2+4a-2)
Y = (a+1)(a^4-14a^3+24a^2-14a+1)
Z = (a-2)(a^4+10a^3-12a^2+4a-2)(a+1)(a^4-14a^3+24a^2-14a+1)
W = (2a-1)(2a^4-4a^3+12a^2-10a-1)
a is arbitrary.

Case : m=3

X = (a^4+a^3+6a^2-14a+7)(a^12+48a^11-789a^10+3406a^9-6876a^8+7083a^7-3396a^6+36a^5+1062a^4-725a^3+156a^2-6a+1)
Y = (a^4-5a^3+15a^2-5a+1)(a^12-60a^11-195a^10+1624a^9-3312a^8+3357a^7-2829a^6+3357a^5-3312a^4+1624a^3-195a^2-60a+1)
Z = (a^4+a^3+6a^2-14a+7)(a^12+48a^11-789a^10+3406a^9-6876a^8+7083a^7-3396a^6+36a^5+1062a^4-725a^3+156a^2-6a+1)
    (a^4-5a^3+15a^2-5a+1)(a^12-60a^11-195a^10+1624a^9-3312a^8+3357a^7-2829a^6+3357a^5-3312a^4+1624a^3-195a^2-60a+1)
W = (7a^4-14a^3+6a^2+a+1)(a^12-6a^11+156a^10-725a^9+1062a^8+36a^7-3396a^6+7083a^5-6876a^4+3406a^3-789a^2+48a+1)

Similarly, we can obtain infinitely many parametric solutions for equation (1).
     
Q.E.D.　
 
           
       

3.Reference

[1]. A. Bremner, AJAI CHOUDHRY and M. Ulas, CONSTRUCTIONS OF DIAGONAL QUARTIC AND SEXTIC
SURFACES WITH INFINITELY MANY RATIONAL POINTS, arxiv.org/pdf/1402.4583, 2014